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Nearly free curves and arrangements: a vector bundle point of view

Part of: (Co)homology theory Discrete geometry Singularities

Published online by Cambridge University Press:  06 September 2019

SIMONE MARCHESI  and
JEAN VALLÈS
SIMONE MARCHESI
Affiliation:
IMECC - UNICAMP, Rua Sergio Buarque de Holanda 651, 13083-859 Campinas, SP, Brazil. e-mail: marchesi@ime.unicamp.br
JEAN VALLÈS
Affiliation:
Université de Pau et des Pays de l’Adour LMAP-UMR CNRS 5142 Avenue de l’Université - BP 576 - 64012 PAU Cedex - France. e-mail: jean.valles@univ-pau.fr
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Abstract

Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i.e. a line arrangement) its freeness depends solely on its combinatorics, but this has only been proved for sets of up to 12 lines. In looking for a counter-example to Terao’s conjecture, the nearly free curves introduced by Dimca and Sticlaru arise naturally. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non-zero section that vanishes on one single point, P say, called the jumping point, and that this characterises the bundle. We then give a precise description of the behaviour of P. Based on detailed examples we then show that the position of P relative to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics.

MSC classification

Primary: 52C35: Arrangements of points, flats, hyperplanes
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 32S22: Relations with arrangements of hyperplanes
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 1 , January 2021 , pp. 51 - 74
Copyright
© Cambridge Philosophical Society 2019

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References

REFERENCES

Abe, T. and Dimca, A.. On the splitting types of bundles of logarithmic vector fields along plane curves. Arxiv:1706.05146.Google Scholar
Barth, W.. Some properties of Stahle rank-2 vector bundles on Pn. Math. Ann. 226 (1977), 125–150.CrossRefGoogle Scholar
Dimca, A. and Sticlaru, G.. Nearly free divisors and rational cuspidal curves. Arxiv:1505.00666.Google Scholar
Elencwajg, G. and Forster, O.. Bounding cohomology groups of vector bundles on Pn. Math. Ann. 246 (1979), no. 3, 251–270.Google Scholar
Faenzi, D. and Vallès, J.. Logarithmic bundles and Line arrangements, an approach via the standard construction. J. Lond. Math. Soc. (2) 90 (2014), no. 3, 675–694.Google Scholar
Grayson, D. R. and Stillman, M. E.. Macaulay, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.Google Scholar
Okonek, C., Schneider, M. and Spindler, H.. Vector bundles on complex projective spaces. Progr. Math., 3 (1980). (Birkhäuser, Boston, Mass).Google Scholar
Orlik, P. and Terao, H.. Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften, 300 (1992). (Springer-Verlag, Berlin).CrossRefGoogle Scholar
Saito, K.. Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 27 (1980), no. 2, 265–291.Google Scholar
Terao, H.. Arrangements of hyperplanes and their freeness. I.J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 27 (1980), no. 2, 293–312.Google Scholar
Vallès, J.. Free divisors in a pencil of curves. Journal of Singularities 11 (2015), 190–197.Google Scholar
Vallès, J.. Erratum: free divisors in a pencil of curves. Journal of Singularities 14 (2016), 1–2.Google Scholar
Wakefield, M. and Yuzvinsky, S.. Derivations of an effective divisor on the complex projective line. Trans. Amer. Math. Soc. 359 (2007), no. 9, 4389–4403.Google Scholar
Ziegler, G.. Multiarrangements of hyperplanes and their freeness. Singularities (Iowa City, IA, 1986), 345–359. Contemp. Math. 90 (1989), Amer. Math. Soc. (Providence, RI).CrossRefGoogle Scholar